Question

select the correct answer.
a farmers silo is a cone on top of a 100 foot - tall cylinder. the height of the cone is three fourths the radius.
what is the maximum radius of the silo (in feet) if there are 6,000 square feet of material?
9.5 feet
8.5 feet
9 feet
8 feet

Explanation:

Step1: Calculate the surface area of the cylinder

The formula for the lateral surface area of a cylinder is \(S_{cylinder}=2\pi rh\), where \(h = 100\) feet. So \(S_{cylinder}=2\pi r\times100=200\pi r\).

Step2: Calculate the slant height of the cone

The height of the cone \(h_{cone}=\frac{3}{4}r\). Using the Pythagorean theorem for the slant height \(l\) of the cone (\(l=\sqrt{r^{2}+h_{cone}^{2}}\)), we substitute \(h_{cone}=\frac{3}{4}r\) and get \(l=\sqrt{r^{2}+(\frac{3}{4}r)^{2}}=\sqrt{r^{2}+\frac{9}{16}r^{2}}=\sqrt{\frac{16r^{2}+9r^{2}}{16}}=\sqrt{\frac{25r^{2}}{16}}=\frac{5}{4}r\).

Step3: Calculate the lateral surface area of the cone

The formula for the lateral surface area of a cone is \(S_{cone}=\pi rl\). Substituting \(l = \frac{5}{4}r\), we get \(S_{cone}=\pi r\times\frac{5}{4}r=\frac{5}{4}\pi r^{2}\).

Step4: Set up the equation for the total surface area

The total surface area \(S = S_{cylinder}+S_{cone}\). Given \(S = 6000\), we have \(200\pi r+\frac{5}{4}\pi r^{2}=6000\). Multiply through by \(\frac{4}{\pi}\) to get \(800r + 5r^{2}=\frac{24000}{\pi}\approx\frac{24000}{3.14}\approx7643.31\). So \(5r^{2}+800r - 7643.31=0\). Using the quadratic formula \(r=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(a = 5\), \(b = 800\), \(c=- 7643.31\). \(r=\frac{-800\pm\sqrt{800^{2}-4\times5\times(-7643.31)}}{2\times5}=\frac{-800\pm\sqrt{640000 + 152866.2}}{10}=\frac{-800\pm\sqrt{792866.2}}{10}=\frac{-800\pm890.43}{10}\). We take the positive root \(r=\frac{- 800 + 890.43}{10}\approx9.0\)

Answer:

9 feet